CALT-TH-2014-145

Stable Bound States of Asymmetric Dark Matter Mark B. Wise and Yue Zhang

arXiv:1407.4121v1 [hep-ph] 15 Jul 2014

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 The simplest renormalizable effective field theories with asymmetric dark matter bound states contain two additional gauge singlet fields one being the dark matter and the other a mediator particle that the dark matter annihilates into. We examine the physics of one such model with a Dirac fermion as the dark matter. For a range of parameters the Yukawa coupling of the dark matter to the mediator gives rise to stable asymmetric dark matter bound states. We derive properties of the bound states including nuggets formed from N 1 dark matter particles. We also consider the formation of bound states in the early universe and direct detection of dark matter bound states. Many of our results also hold for symmetric dark matter.

I.

INTRODUCTION

An attractive idea for the origin of the cosmological dark matter (DM) density relies on a primordial DM asymmetry that prevents the near complete annihilation of the DM particles with their anti-particles resulting in the observed relic density. DM that arises from this mechanism is called asymmetric dark matter (ADM) [1]. If DM is asymmetric then the universe today is composed of DM particles with their antiparticles absent. An interesting possibility that can impact the properties of DM relevant for direct and indirect detections and influence its distribution in galaxies is the presence of stable bound states of DM particles. For ADM these bound states involve DM particles (but not DM anti-particles). The cosmology of ADM bound states have been explored previously in various contexts, such as the dark atom models with two species of ADM [2–5], and strongly interacting non-abelian hidden sector models [6–8]. In the Standard Model (SM) there is no particle that can be the DM. Minimality can be a useful guiding principle when considering speculative extensions of the SM. The purpose of this paper is to study one of the two minimal renormalizable extensions of the SM with stable ADM bound states. We will not discuss the mechanism that generates the primordial DM asymmetry. Rather we focus on the low energy effective theory well below the scale where the primordial asymmetry is generated. In the model we study, the DM is a Dirac fermion χ with no standard model quantum numbers. To facilitate annihilation of χ particles with their anti-particles χ ¯ in the early universe, we introduce a Yukawa-coupling of χ to a real scalar field φ lighter than the DM. Unlike U (1) gauge boson exchange1 , scalar exchange is always attractive among particles (or antiparticles), and so for a range of model parameters stable ADM bound states occur2 in the model we consider. The new scalar φ mixes with the Higgs boson, allowing it to decay to SM particles, and mediate interactions between ADM and SM particles that lead to a possible signal in the direct detection experiments. We discuss the spectrum of non-relativistic ADM bound states from two-body to multi-particle bound states, and the production of bound states in the early universe. We find regions of parameter space where most of the DM particles do not reside in bound states, and regions where most of the DM does. The border between these two regions occurs (roughly 1 2

DM interacting via dark U (1) gauge boson exchange was considered in for example Refs. [9, 10]. Having additional scalars aggravates the hierarchy problem of the standard model but in this case since the scalar is related to the DM density there may be environmental reasons for it being light compared with the Planck or GUT scales.

2 speaking) when the force mediator φ mass is comparable to the largest binding energy of the two body bound states.3 The low energy effective theory of ADM we consider must be (approximately) invariant under global U (1)χ transformations, χ → eiα χ. Like the ADM density the baryon density of the universe may arise from a primordial asymmetry, in that case in B − L. For the baryon asymmetry of the universe, the (approximate) conservation of B − L in the low energy effective theory, i.e., the SM, below the scale where the primordial asymmetry is generated is a consequence of gauge invariance and the particle content, since no renormalizable interactions between SM particles violate B − L.4 Naively this is not true for the ADM model we consider, global U (1)χ invariance is not an automatic consequence of the gauge symmetries and particle content. For example, while a Dirac mass term χχ ¯ preserves χ number a Majorana mass term χ ¯c χ does not. However, there could be an unbroken discrete gauge subgroup of U (1)χ that forbids the Majorana mass term resulting in a low energy effective theory, where the gauge symmetries are enough to ensure the renormalizable couplings in the low energy theory are invariant under U (1)χ . Even in models where the DM is not asymmetric, some of the results of the paper may be applicable. Scalar exchange could gives rise to stable bound states of DM particles and stable bound states of anti-DM particles. This will have an impact in cosmology and for direct detection. For a range of parameters, such bound states occur in for example [11–14]. II.

THE SIMPLEST FERMIONIC ADM EFT WITH BOUND STATES

The most minimal renormalizable effective theory for ADM contains complex scalar DM that annihilates via the Higgs portal in the early universe. However, it is not difficult to show that in this case there is not a robust region of parameter space where non-relativistic ADM bound states exist. Demanding the existence of stable ADM bound states in a renormalizable model requires one more degree of freedom than this minimal model. We focus on a renormalizable model with Dirac fermionic ADM χ and a real scalar mediator φ, 1 1 1 1 L = iχ ¯ 6 ∂χ − mχ χχ ¯ − χ(g ¯ χ + ig5 γ5 )χφ + (∂φ)2 − λ4φ φ4 − λ3φ φ3 − m2φ φ2 2 24 6 2 1 2 † 2 † 2 (1) − µφh φ(H H − v /2) − λφh φ (H H − v /2) − V (H) , 2 where H is the SM Higgs doublet, and V (H) is the usual Higgs √ potential. The vacuum expectation value for the neutral component of the Higgs doublet is v/ 2, and v = 246 GeV. We have shifted the scalar field φ so that it has no vacuum expectation value. After electroweak symmetry breaking, φ picks up a small mixing with the SM Higgs boson h, thus allowing it to decay into SM particles, and mediate the DM interaction to the SM. This simple model can already have a very complicated spectrum of bound states with important implications for cosmology and direct detection. With the above interactions, we calculate the cross section for χχ ¯ annihilation into the lighter mediator during thermal freeze out. In the region of parameter space with bound states mφ is much less than mχ . Neglecting mφ /mχ , µφh /mχ , λ3φ /mχ , and expanding to order v 2 the annihilation cross section is, 3π 2 2 2π π 2 2 hσvianni = α v + 2 αχ αI + α v , (2) 2m2 χ m 6m2 I 3

4

The other minimal renormalizable extension of the standard model with stable ADM bound states has scalar dark matter. Violation of this global symmetry occurs at dimension five through the operators that are responsible for neutrino masses.

3

m Χ = 1 TeV

m Χ = 100 GeV

1.000 0.500

m Χ = 10 GeV

1.000 BBN&LUX excl. minimal model only

1.000 BBN&LUX excl. minimal model only

0.500

1 GeV

10 GeV

0.100

100 MeV

0.100 1 GeV

0.100 100 MeV

10 MeV

ΑΧ

0.050

ΑΧ

0.050

ΑΧ

0.050 100 MeV

10 MeV

0.010

1 MeV

0.010 10 MeV

0.010 1 MeV

0.005

0.005 1 MeV

0.001 0.001

No bound state 0.01

0.1 1 mΦ HGeVL

10

100

BBN&LUX excl. minimal model only

0.500

100 keV

No bound state

0.005

No bound state

100 keV

0.001 0.001

0.01

0.1 mΦ HGeVL

1

10

10 keV

0.001 0.001

0.01

0.1 mΦ HGeVL

1

3

FIG. 1: Ground state binding energy (in blue curves) of two-body DM bound state in the αχ − mφ space, for DM mass equal to 1 TeV (left), 100 GeV (middle) and 10 GeV (right), respectively. No two-body bound state exists in the dark blue region, and only one (1s) bound state exists in the lighter blue region. The yellow region is not consistent with BBN and LUX constraints in the minimal model described by Eq. (1). The black dashed curves are lower bound on αχ from having a large enough annihilation rate for the ADM, in the case αI = 0 (upper) and αI = αχ , respectively.

where v is the relative velocity between χ, χ, ¯ given by mχ v 2 ' 3T , and αχ ≡ gχ2 /(4π), αI ≡ g52 /(4π). We work to lowest order in perturbation theory neglecting the Sommerfeld enhancement factor which is not very important at freeze out [11, 15]. Asymmetric DM needs a larger annihilation rate than the WIMP, hσv(T ' mχ /26)ianni > 3 × 1026 cm3 /s [16]. With this condition, the symmetric component of DM can annihilate efficiently, and today’s relic density is dictated by the initial DM asymmetry. For the χχ ¯ annihilation, the contribution from the coupling gχ is velocity suppressed but not the contribution from interference between gχ and g5 . For non-relativistic interactions between DM particles the influence of g5 that is suppressed. In Fig. 1 the dashed lines are the limits on αχ assuming αI = 0 (upper dashed line) and αχ = αI (lower dashed lines) that arise from demanding that enough annihilations take place. In order to be consistent with Big-Bang Nucleosynthesis (BBN), φ has to decay before a second or so. In this model, φ decays to SM particles via mixing with the Higgs boson h, so its decays are similar to those of the Higgs boson [17]. For mφ below twice of the pion mass, its decay rate is !3/2 !3/2 G m2 m GF m2µ mφ 4m2µ 4m2e F e φ √ √ Γφ = 1− 2 Θ(mφ − 2me ) + 1− 2 Θ(mφ − 2mµ ) 4 2π mφ mφ 4 2π " #2 2 2 µ2 v 2 GF α2 m3φ 2 −1 φh √ + A1/2 + 3 A1/2 , (3) A1 + 2 × 3 3 3 3 m4h 128 2π where the second row is for φ → γγ decay and we have taken into account of the W -loop (A1 term) and the heavy quark loops (A1/2 term) from t, b, c, and A1 = −7, A1/2 = 4/3 [18]. The φ → µ+ µ− decay dominates above the two-muon threshold. For heavier φ, hadronic decay channels open, and the dimuon branching ratio reduces to around 10% [19], for φ mass up to a few GeV. The φ − h mixing can also mediate the interaction between DM and SM particles, and lead to

4

2 ´ 10-5

@ΜΦh vmh 2 DMax

1 ´ 10-5 5 ´ 10-6 2 ´ 10-6 1 ´ 10-6 5 ´ 10-7 2 ´ 10-7

10

20

50

100 200 m Χ HGeVL

500 1000 2000

FIG. 2: Upper bound on the φ − h mixing parameter µφh v/m2h from dark matter direct detection (LUX), assuming all DM are free particles today. The solid and dashed curves corresponds to αχ = 0.1 and 0.01, respectively. We fix mφ = 500 MeV.

DM direct detection signals. The direct detection cross section for χ scattering on a nucleon is [20] σSI =

4αχ f 2 m4N µ2φh m4h m4φ

,

(4)

assuming m2φ |q 2 | where q is the momentum transfer, and f ' 0.35 [21]. Fig. 2 shows the constraints on the φ − h mixing from the DM direct detection by the LUX collaboration [22] for given φ mass and αχ . We find that combining the LUX and BBN constraints generically requires the φ → µ+ µ− decay channel to be open, i.e., mφ & 210 MeV. The regions of parameter space that are excluded by direct detection and consistent with BBN are shaded yellow in Fig. 1. This eliminates a large part of the available parameter space. More complicated models with an enlarged dark sector may evade this constraint. See the appendix for an example. A.

Two-body Bound States

The exchange of a scalar field φ gives an attractive force among the ADM particles. For sufficiently light φ, most of the ADM in the universe could reside in bound states. Because the φ force is always attractive, multi-particle ADM bound states (N ≥ 2) are expected. In this section, we discuss two-body bound states. The states containing more than two DM will be discussed in the following subsection. For two-DM-particle system, the non-relativistic Hamiltonian describing two DM interaction is H = −

∇2cm ∇2 αχ −mφ r − − e + Hint , 4mχ mχ r

(5)

where xcm is the center of mass coordinate, and r is the relative position of two χ’s inside the bound state. Hint is the interaction for on-shell φ creation/annihilation, which controls transition rates, that involve φ particle emission and absorption. The other terms in Eq. (5) control the spectra and wave functions of the two-body bound states in the non-relativistic limit. We restrict our attention

5 to the region of parameter space where αχ . 0.1 which insures that the two-particle bound states are non-relativistic. The non-relativistic two-body bound state problem with a Yukawa potential has been solved numerically in [23] (see also [24]). In order for a bound state to exist, the screening length (∼ 1/mφ ) must be large enough compared to the size of the corresponding wave function (∼ the Bohr radius). The condition for having at least one bound state (the 1s state) is αχ mχ > 0.8399 . 2mφ

(6)

In Fig. 1, we plotted contours of fixed 1s state binding energy as thin blue curves in the αχ versus mφ plane, for fixed DM mass in each panel. The dark blue region does not satisfy the bound state condition Eq. (6) because φ is too heavy. In the light blue region, 1s state is the only existing bound state. Outside these regions, for mφ αχ mχ , the φ force is very close to Coulombic, and the solution approaches to the hydrogen-like state, with a Bohr radius a0 =

2 . αχ mχ

(7)

In this case, the ground state wave function and binding energy are, approximately, 1 −3/2 ψ0 (r) = √ a0 e−r/a0 , π

BE 0 =

αχ2 mχ . 4

(8)

A two-particle bound state Bi can decay to one with greater binding energy Bf by real or virtual φ emission. Real φ emission is kinematically allowed when the the difference in binding energies ∆BE between the final and initial states is greater than the mass of the φ. Also bound states can be formed by scattering two χ’s and emitting either a real or virtual φ and dissociated by scattering the bound state with a φ. The couplings of the φ to the SM particles are restricted to be very small by current direct detection bounds so we concentrate on real φ emission and absorption in these processes. Also for simplicity we only consider the s-wave bound states. They have the spins of the χ’s combined into a total spin zero state. The interaction Hamiltonian that is needed for these calculations is h r r i Hint = gr φ xcm + + φ xcm − . (9) 2 2 Our calculations are valid as long as the χ’s are non-relativistic but the φ can be relativistic. Excited bound states can decay to lower bound states by φ emission, Γ(Bi → Bf + φ) = 8αχ k|Gif (k)|2 ,

(10) q where k = ∆BE 2 − m2φ is the magnitude of the φ momentum and the transition form factor is Z Gif (k) =

d3 re−k·r/2 ψf∗ (r)ψi (r) .

(11)

We will need, for the cosmology discussion in the forthcoming section, the cross sections for the formation and dissociation of the two-body states via real φ emission and absorption (see Fig. 3). We first consider the formation process in the center of mass frame, χ(p) + χ(−p) → Bi + φ. The cross section is expressed in terms of the form factor. Z Fi (p, k) = d3 re−ip·r ψi (r)(eik·r/2 + e−ik·r/2 ) , (12)

6

B

B

FIG. 3: Feynman diagrams for two-body ADM bound state formation (dissociation) with an on-shell φ emission (absorption).

where ψi is the spatial wave function for the i-th s-wave bound state and k is the three momentum of the outgoing φ. In the cases considered here, we can use the dipole approximation, neglecting the φ three momentum k. In the case of the ground state i = 0 approximating the wave function by the Coulombic one in Eq. (8), F0 (p, 0) =

√ 3 16 πa02 1 + a20 |p|2

2 .

(13)

Using the dipole approximation, in the center of mass frame, σ(χ(p) + χ(−p) → Bi + φ)v =

αχ k|Fi (p, 0)|2 , 2

where energy conservation fixes the magnitude of the final state φ momentum to be, q k = (BEi + p2 /mχ )2 − m2φ .

(14)

(15)

It is evident from the above formula that as long as the χ’s are non-relativistic, the dipole approximation is reasonable. Similarly the cross section for the dissociation process in the lab frame, Bi (0) + φ(k) → χ + χ is, σ(Bi (0) + φ(k) → χ + χ)v =

αχ mχ p|Fi (p, 0)|2 , Ek

(16)

where p is the magnitude of the relative momentum of the final stateqχ’s, i.e. p = (p1 − p2 )/2 with

p1 and p2 are the three momenta of the two final state χ’s. Ek = k 2 + m2φ is the energy of the incoming φ. Energy conservation determines the magnitude of the relative χ momentum to be, q p = mχ (Ek − BEi ) , (17) where BEi is the binding energy of the two body bound state Bi . If some of the DM is in the two-body bound ground state today the direct detection predictions may be modified. The momentum transfer associated with direct detection is less than 1/a0 so under most circumstances the scattering is coherent, the form factor suppression negligible and the cross section is just 4 times the single χ cross section in Eq. (4). An exception occurs when the two body scattering length is very large. At zero momentum the (spin zero) χ − χ scattering cross section σ(χχ → χχ) = 4πa2 , where a is the spin zero scattering length. If there is a two body bound state (or resonance) very near threashold (i.e. zero binding energy) then the scattering

7 length a is very large5 . In this case there are universal properties at low momentum for few body χ systems. This is familiar from the effective range expansion for two body nuclear physics processes (e.g. n + p → d + γ) and was applied to DM direct detection in [25]. B.

Dark Matter Nuggets

As mentioned above, the φ force among DM particles is always attractive. Therefore, for small enough φ mass we expect DM to have N > 2 particle bound states (nuggets). For sufficiently large N , we assume the DM nugget can be described as degenerate Fermi gas. Undoubtably this is a gross simplification of the dynamics. One expects more complicated phenomena like pair formation [26] to occur and impact the equation of state for the DM. However, our main purpose here is not a quantitative analysis of the spectrum and properties of multi-particle DM bound states but rather to argue that such states exist and that they are probably small enough that in direct detection experiments the scattering is coherent and gives rise to a cross section that grows as N 2 . This will also affect the capture rate for DM by neutron stars [27, 28]. We first discuss this multi-particle bound state problem using a heuristic approach where the DM density is constant and then a more quantitative approach that relies on hydrostatic equilibrium. In expressions for physical quantities the scaling with N , mχ and αχ is the same in the heuristic and hydrostatic equilibrium approaches. We restrict our attention to the non-relativistic regime so we can neglect relativistic corrections to the potential energy, for example from the Darwin term. Also we will find that the size of the state decreases with N so that we can neglect mφ replacing the Yukawa potential by a Coulomb potential. 1.

Heuristic Approach

Assuming a constant density of χ particles filling the momentum levels up to the Fermi momentum pF the number density of χ particles is n=

p3F . 3π 2

(18)

For a spherical volume of radius R, the total number of particles is N = 4p3F R3 /(9π). The kinetic energy KE, and potential energy P E, of the χ particles expressed as a function or the total number of particles N and the radius R are 5/3 3N 2 αχ 2π 2/3 5/3 9 KE = N , P E = − . (19) 15R2 mχ 4 5R Minimizing the total energy E = KE + P E with respect to R at fixed N determines the radius to be, 2/3 1 9π R= . (20) 1/3 4 N αχ mχ Notice that the volume of the nugget decreases as 1/N and if mφ a0 1, then we also have for large N that mφ R 1. Hence as long as two body bound states exist the Yukawa potential can be treated as Coulombic for nuggets. 5

This is the origin of the enhancements in [11], [13] and [14].

8 At the value of R in Eq. (20) the kinetic and potential energies become, KE ' 0.08 αχ2 mχ N 7/3 , P E ' −0.16 αχ2 mχ N 7/3 .

(21)

For large N the nuggets have a binding energy that is of order N 4/3 times the binding energy of N/2 two body bound states BE0 . There are a number of conditions that must be satisfied to apply even the crude approximations we have made. Firstly as N increases the Fermi momentum increases and the system eventually becomes relativistic. Demanding that pF /mχ 1 implies that N

9π 4

1/2

αχ−3/2

' 2.7

1 αχ

3/2 .

(22)

Determining the properties of the nugget using partly classical methods will be valid if pF R 1. This implies that −1/3 9π ' 0.52 . (23) N 4 In the presence of a background number density for χ’s, the Yukawa coupling of the scalar induces a density dependent tadpole and for large enough R a scalar expectation value. However, neglecting mφ and λ3φ and treating λ4φ as order unity, p we find that these effects are subdominant compared to those we have included, provided N 1/αχ . 2.

Hydrostatic Equilibrium

For a degenerate Fermi gas the equation of state relating the pressure density p to the number density n is p = Kn5/3 ,

(24)

where K = 5−1 32/3 π 4/3 m−1 χ , and n is the number density of χ particles. For a stable solution, the Fermi pressure is balanced by the attractive φ force among the particles. When 1/mφ is larger than the size of the nugget, the attractive force is Coulomb-like. In this case, the hydrostatic equilibrium equation is 1 d r2 dp = −4παχ n . (25) r2 dr n dr Together with the above equation of state, Eq. (24), this equation can be solved for N DM particles and it has a finite-size solution, with R=

(3π)2/3 2−7/3 2 0 4.5 (ξ1 |θ (ξ1 )|)1/3 · ξ1 ' 1/3 , 1/3 N αχ mχ N αχ mχ

(26)

where θ(ξ) is the solution to the Lane-Emden equation with index n = 3/2, and ξ1 = 3.65, ξ12 |θ0 (ξ1 )| = 2.71 [29] . The Fermi momentum near the center of the nugget is (pF )c =

22/3 N 2/3 αχ mχ ' 0.1N 2/3 αχ mχ . 31/3 π 4/3 (ξ12 |θ0 (ξ1 )|)2/3

For this description to apply, there are consistency conditions.

(27)

9 • Non-relativistic condition: (pF )c mχ requires N (0.1αχ )−3/2 . For αχ ≤ 0.1, the right hand side is of order 103 or larger. • Classical description: (pF )c R 1 requires N 6.4 . • Long range force condition: mφ R 1 requires mφ N 1/3 αχ mχ /4.5 . For large N nuggets are smaller than the the two-body bound ground state, while the momentum of DM inside the nugget is much larger than the two-body case. Therefore, as long as the two-body bound state exists, the screening effect due to mφ can be neglected. Within the degenerate Fermi gas picture, for very large N , a non-relativistic description is no longer valid. Other interactions we have not included become important in the analysis of such systems. However, it seems plausible that relativistic bound states exist, and even black holes. If most of the DM ends up in black holes, then they must be cosmologically stable. As we noted before, since the size of DM nuggets shrinks with N we expect their direct detection scattering cross section to be coherent and be proportional to N 2 . III.

COSMOLOGY

In this section, we study the formation of bound states in the early universe. In general, there are two stages when bound states may form most efficiently: 1) shortly after the DM freeze out when the ADM number density is still high, 2) at a later stage where structure growth has gone non-linear and the DM density can be locally large. We first perform a calculation of two-body bound state formation in stage 1), taking into account two competing processes, formation χχ → Bφ, and dissociation Bφ → χχ. The fraction of DM in bound states depends on the interplay between these two rates and the Hubble parameter. Our goal here is to understand two-body bound state formation. To calculate the formation of bound states with more than two particles, we need to know the binding energies and wave functions of those states. For convenience, in this section we fix µφh v/m2h = 10−6 and take mφ > 2mµ and less than a few GeV. These values are consistent with constraints from DM direct detection. In this region of parameter space, φ decay is dominated by hadronic and the two muon-final state. Neglecting threshold effects, the two-muon contribution gives the bound, 1 GeV −4 τφ . 10 sec . (28) mφ Recall that then the universe is 10−4 second old, its temperature is ∼ 100 MeV. When t < 1/τφ in the early universe there is a plasma of φ particles which couples to the DM. In order to calculate the averaged bound state formation and dissociation rates in this plasma, we need to know the energy/momentum distributions of χ and φ. At very high temperature, the mediator φ was in thermal equilibrium with SM fermions via the Higgs boson exchange, φφ ↔ f f¯. For λφh ∼ O(1) and mφ . 1 GeV, such interactions freeze out at temperature equal to 1 GeV or so, slightly below the charm quark threshold. Afterwards, φ can only remain in chemical equilibrium with itself through the 2 ↔ 3 scattering φφ ↔ φφφ, with the λ3φ , λ4φ couplings. This allows it to have its own temperature Tφ , which satisfies Tφ ' Tγ

1, [g∗ (Tγ )/g∗ (1 GeV)]1/3 ,

Tγ > 1 GeV Tγ < 1 GeV

(29)

10 where Tγ is the photon temperature and g∗ (Tγ ) is the number of relativistic degrees of freedom in the SM at Tγ . When the temperature falls below mφ , the φ number density becomes Boltzmann suppressed and the 2 ↔ 3 scattering process freezes out, at temperature around an order of magnitude below mφ . Another important way to deplete the φ’s is decay. For simplicity, we restrict ourselves to the region of parameter space where φ decay happens before or around the freeze out, and in this case the phase-space distribution of φ is fφ (E) ' e−E/Tφ e−1/(2Hτφ ) .

(30)

Second, after the anti-DM χ ¯ are efficiently depleted (T . mχ /30), the remaining ADM component χ can stay in kinetic equilibrium with φ, via the elastic scattering χφ → χφ. For the range of parameters we study, this rate is always larger than the Hubble rate, until φ’s decay away. In this case, the phase space distribution of DM χ is Tγ3 ΩDM mp −p2 /(2mχ Tφ ) 5/2 −1/2 η fχ (p) = Ce , C=2 π ζ(3) , (31) Ωb mχ (mχ Tφ )3/2 where η ' 6 × 10−10 is the ratio of baryon to photon number in the universe, and mp is the mass of the proton. We will use the φ and χ distributions described above to calculate the thermal averaged rates. For the formation rate Γf orm per particle, we do a thermal integral of the cross section Eq. (14) over the incoming χ momentum. The thermally averaged dissociation rate Γdiss per particle is obtained by integrating Eq. (16) over the incoming φ energy. When both Γf orm and Γdiss are larger than the P Hubble rate and only two-body bound states exist, the dark ionization fraction, Xd ≡ nχ /(2 i nBi + nχ ), (the index i goes over all possible two-body bound states) satisfies the dark sector counterpart of the Saha equation6 1 − Xd 8ζ(3) = √ η 2 π Xd

5.4 GeV mχ

Tφ mχ

3/2 X

eBEi /T .

(32)

i

The large number of φ’s in the plasma implies that Γdiss Γf orm until the time of φ decay. In the end, how much two-body bound states are formed is determined by the comparison of formation rate and Hubble rate at that time. For simplicity below, we only discuss the formation and dissociation of the ground state. However, kinematically, excited states are harder to form and easier to destroy. Therefore, we expect most of the two-body bound state formation to occur in the ground state. The parameter space can be divided into two regimes. • Case A: mφ BE0 . In this case, the mediator mass is much larger than the binding energy, and the formation process χχ → Bφ cannot happen unless the two DM are energetic enough. In other words, for DM in kinetic equilibrium, the temperature of the universe must be large enough. When the temperature falls below a threshold Tth = (mφ − BE0 )/3 ' mφ /3, the formation rate becomes exponentially suppressed. In contrast, with a plasma of φ the dissociation rate does not shut off until φ eventually decays, which happens at a temperature much lower than Tth . This feature is shown as the left panel of Fig. 4, where when Γf orm falls below H, Γdiss is still much larger than the Hubble rate H. In this case, any bound states that were formed will 6

A similar equation could be derived in the nugget case.

11 Case B: m Χ = 1 TeV, Α Χ =0.1, mΦ =300 MeV 1012

104

104

10-4

10-4

-12

10

GΦB® ΧΧ

Rates HGeVL

Rates HGeVL

Case A: m Χ = 100 GeV, Α Χ =0.05, mΦ =300 MeV 1012

G ΧΧ®BΦ

10-20

H

-28

10

10-36 m Χ 26

10-12

G ΧΧ®BΦ

10-20

H

10 mΦ

10 MeV

1 MeV

GΦB® ΧΧ

-28

10-36 m Χ 26

mΦ

T

10 MeV

1 MeV

T

FIG. 4: Temperature dependence of the two-body ADM bound state formation (red), dissociation (blue) rates in together with the Hubble expansion rate (black). In both cases, we set µφh v/m2h = 10−6 such that both DM direct detection and BBN bounds can be satisfied. m Χ = 200 GeV, Α Χ =0.08, mΦ =500 MeV 10

12

Rates HGeVL

104

GΦB® ΧΧ

10-4 10-12

G ΧΧ®BΦ

10-20 10

H

-28

10-36 m Χ 26

mΦ

10 MeV

1 MeV

T

FIG. 5: Same as Fig. 4, but the parameters are chosen such that the critical condition discussed above Eq. (35) is satisfied.

eventually be ionized back to unbounded DM particles. For the same reason, bound state formation when the structure growth becomes non-linear is also suppressed because the DM is more non-relativistic. • Case B: mφ BE0 . In this case, the binding energy release itself is sufficient to produce an on-shell φ. There is no temperature threshold for bound state formation. The thermally averaged formation cross section Eq. (14) can be simplified in the following two regimes 4παχ6 mχ /(27Tφ3 ), Tφ BE0 hσ(χ + χ → B0 + φ)vi ' (33) 2 256π/mχ , Tφ BE0 Because nχ ∼ Tφ3 , the formation rate remains constant when Tφ BE0 , and scale as ∼ Tφ3 (decreases faster than Hubble rate) when Tφ BE0 . Hence there is a lower bound on the

12

m Χ = 1 TeV

m Χ = 100 GeV

1.000

m Χ = 10 GeV

1.000

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m

E0 =B Φ

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0

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E =B mΦ

0.500

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0.005

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3

FIG. 6: Same parameter space as Fig. 1. The thick red curves represent where the critical condition like Fig. 5 is reached. The dot-dashed lines are where mφ = BE0 is satisfied.

coupling constant αχ for the formation rate to ever be larger than the Hubble rate mχ mχ ' 5 , αχ & p 10 GeV ηMpl mp

(34)

where Mpl = 1.2 × 1019 GeV is the Planck scale. If this condition is satisfied, the formation rate can remain greater than the Hubble rate for a long time. In contrast, there will be a threshold for dissociation. At mφ < T < EB , there is a suppression in the number of φ in the plasma that are energetic enough to ionize the bound state. Moreover, there is a sharper suppression when φ begins to decay. This allows us to have a picture where the ground state formation process is still active (Γf orm > H) when dissociation is suppressed (Γdiss < H), as shown in the right panel of Fig. 4. In this regime, the two-body ground state can efficiently form. In this regime, the formation of more than two-body bound states is also expected to be efficient. • The critical case. In the region that interpolates between the two above limiting cases, for given mφ , we find it is always possible to arrange the parameters such that both formation and dissociation freeze out at similar temperature Tc , where Γdiss (Tc ) = H(Tc ), but Γf orm (Tc ) is somewhat below H(Tc ), as shown in Fig. 5. Below Tc , the dissociation rate is more suppressed because of φ decay. In this case, the fraction of DM particles that finally end up in the two-body ground state is roughly, P '

Γf orm (Tc ) . H

(35)

Since P is smaller than one, we expect the formation of more than two-body bound states to be further suppressed. We summarize the results in the αχ versus mφ parameter space in Fig. 6. The thick red curves are where the above critical condition is satisfied. Below the thick red curves, almost all the DM ends up as unbound χ particles. Above thick red curves, most of the DM resides in bound states. These curves lie at mφ somewhat above the binding energy BE0 because of the kinetic energy of the χ particles (see Eq. (15)). Note the thick red curves bend up towards the mφ = BE0 line near

13 the two muon threshold for φ decay. This occurs because the φ decay rate is suppressed which postpones the suppression of dissociation rate, and a larger value of αχ is needed to reach the critical point. The force between any pair of DM particles is attractive and so for mφ a0 1, the binding energy for a bound state with N particles grows faster than linearly with N . That was what we found in the degenerate Fermi gas model where the binding energy grew as N 7/3 . Thus it is likely that for mφ < BE0 there are no thresholds that suppress the formation of non-relativistic bound states with more than two particles. IV.

CONCLUDING REMARKS

We showed that, for a range of parameters, one of the simplest effective theories of asymmetric dark matter has a rich spectrum of bound states. In this paper, we explored some of the features of the spectrum, and the implications for cosmology and dark matter direct detection. We find a region of parameter space where the dark matter in the universe is primarily in bound states. Roughly speaking, this occurs when the binding energy of the two-particle ground state is greater than the mediator mass. However, there are a number of issues that require further examination. For example, the details of the spectrum of the bound states with more than two dark matter particles, i.e., nuggets, and the formation of these multi-particle bound states in the early universe. Without further investigation of these issues, it is even conceivable that for a range of parameters most of the asymmetric dark matter ends up as black holes. In that case, the black holes must have a lifetime longer than the age of the universe. It seems worthwhile to elucidate further the bound state properties and cosmology in this simple model for asymmetric dark matter. Some of the work in this paper is also applicable to dark matter that is not asymmetric. For a range of parameters, scalar exchange could gives rise to stable bound states of dark matter particles and stable bound states of anti dark matter particles. One difference from the asymmetric case is that the values of αχ and αI are constrained to give the correct dark matter relic density. Acknowledgement

We would like to thank Sean Carroll, Clifford Cheung, Francis-Yan Cyr-Racine, Michael RamseyMusolf, David Sanford, Philip Schuster and Natalia Toro for useful discussions and comments on a draft. This work is supported by the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics, and by the DOE Grant DE-SC0011632, and also by a DOE Early Career Award under Grant No. DE-SC0010255. APPENDIX: A MORE NATURAL MODEL

The smallness of µφh in the minimal we consider requires an awkward fine-tuning that is unlikely to have an environment origin. More complicated models can avoid this feature. For example, suppose the dark sector possesses a global dark isospin symmetry SU (2), under which the DM χ is a doublet and the mediator φ is a triplet. The Lagrangian is 1 1 1 1 L = iχ ¯ 6 ∂χ − mχ χχ ¯ − gχ χΦχ ¯ + Tr(∂Φ)2 − λ4φ TrΦ4 − λ3φ TrΦ3 − m2φ TrΦ2 4 24 6 2 1 − λφh TrΦ2 (H † H − v 2 /2) − V (H) , 2

(36)

14 where Φ = σ a φa . In order for the φ to decay, a dark doublet of left-handed fermions ψ is introduced that couples to Φ via the interaction ψ¯c Φψ + h.c.. Note the dark isospin forbids a mass term for ψ. The coupling between Φ and ψ should be large enough for Φ to decay before BBN. In this model, the DM direct detection occurs at the one loop level (see Fig. 7). The cross section is given by Eq. (4), with µφh /m2φ replaced by ∼ gχ λφh /(16π 2 mχ ), which is adequately small even for λφh of order unity.

v

p

h p

FIG. 7: Feynman diagram for direct detection in the SU (2) model.

[1] For a recent review, see K. M. Zurek, Phys. Rept. 537, 91 (2014) [arXiv:1308.0338 [hep-ph]] and references therein. [2] D. E. Kaplan, G. Z. Krnjaic, K. R. Rehermann and C. M. Wells, JCAP 1005, 021 (2010) [arXiv:0909.0753 [hep-ph]]. [3] J. M. Cline, Z. Liu and W. Xue, Phys. Rev. D 85, 101302 (2012) [arXiv:1201.4858 [hep-ph]]. [4] F. -Y. Cyr-Racine and K. Sigurdson, Phys. Rev. D 87, no. 10, 103515 (2013) [arXiv:1209.5752 [astroph.CO]]. [5] J. Fan, A. Katz, L. Randall and M. Reece, Phys. Dark Univ. 2, 139 (2013) [arXiv:1303.1521 [astroph.CO]]. [6] D. S. M. Alves, S. R. Behbahani, P. Schuster and J. G. Wacker, Phys. Lett. B 692, 323 (2010) [arXiv:0903.3945 [hep-ph]]. D. Spier Moreira Alves, S. R. Behbahani, P. Schuster and J. G. Wacker, JHEP 1006, 113 (2010) [arXiv:1003.4729 [hep-ph]]. [7] G. Krnjaic and K. Sigurdson, arXiv:1406.1171 [hep-ph]. [8] W. Detmold, M. McCullough and A. Pochinsky, arXiv:1406.2276 [hep-ph]. W. Detmold, M. McCullough and A. Pochinsky, arXiv:1406.4116 [hep-lat]. [9] L. Ackerman, M. R. Buckley, S. M. Carroll and M. Kamionkowski, Phys. Rev. D 79, 023519 (2009) [arXiv:0810.5126 [hep-ph]]. [10] J. L. Feng, M. Kaplinghat, H. Tu and H. -B. Yu, JCAP 0907, 004 (2009) [arXiv:0905.3039 [hep-ph]]. [11] J. D. March-Russell and S. M. West, Phys. Lett. B 676, 133 (2009) [arXiv:0812.0559 [astro-ph]]. [12] J. L. Feng, M. Kaplinghat and H. -B. Yu, Phys. Rev. Lett. 104, 151301 (2010) [arXiv:0911.0422 [hepph]]. [13] M. R. Buckley and P. J. Fox, Phys. Rev. D 81, 083522 (2010) [arXiv:0911.3898 [hep-ph]]. [14] S. Tulin, H. -B. Yu and K. M. Zurek, Phys. Rev. Lett. 110, no. 11, 111301 (2013) [arXiv:1210.0900 [hep-ph]]. S. Tulin, H. -B. Yu and K. M. Zurek, Phys. Rev. D 87, no. 11, 115007 (2013) [arXiv:1302.3898 [hep-ph]]. [15] N. Arkani-Hamed, D. P. Finkbeiner, T. R. Slatyer and N. Weiner, Phys. Rev. D 79, 015014 (2009) [arXiv:0810.0713 [hep-ph]]. [16] M. Cirelli, N. Fornengo and A. Strumia, Nucl. Phys. B 753, 178 (2006) [hep-ph/0512090]. [17] J. F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, Front. Phys. 80, 1 (2000). [18] M. Carena, I. Low and C. E. M. Wagner, JHEP 1208, 060 (2012) [arXiv:1206.1082 [hep-ph]]. [19] J. D. Clarke, R. Foot and R. R. Volkas, JHEP 1402, 123 (2014) [arXiv:1310.8042 [hep-ph]].

15 [20] M. A. Fedderke, J. -Y. Chen, E. W. Kolb and L. -T. Wang, arXiv:1404.2283 [hep-ph]. [21] J. Giedt, A. W. Thomas and R. D. Young, Phys. Rev. Lett. 103, 201802 (2009) [arXiv:0907.4177 [hep-ph]]. [22] D. S. Akerib et al. [LUX Collaboration], Phys. Rev. Lett. 112, 091303 (2014) [arXiv:1310.8214 [astroph.CO]]. [23] F. J. Rogers, H. C. Graboske, Jr., and D. J. Harwood, Phys. Rev. A 1, 1577 (1970) [24] W. Shepherd, T. M. P. Tait and G. Zaharijas, Phys. Rev. D 79, 055022 (2009) [arXiv:0901.2125 [hepph]]. [25] E. Braaten and H. -W. Hammer, Phys. Rev. D 88, 063511 (2013) [arXiv:1303.4682 [hep-ph]]. R. Laha and E. Braaten, Phys. Rev. D 89, 103510 (2014) [arXiv:1311.6386 [hep-ph]]. [26] R. Shankar, Rev. Mod. Phys. 66, 129 (1994). [27] S. D. McDermott, H. -B. Yu and K. M. Zurek, Phys. Rev. D 85, 023519 (2012) [arXiv:1103.5472 [hep-ph]]. [28] N. F. Bell, A. Melatos and K. Petraki, Phys. Rev. D 87, no. 12, 123507 (2013) [arXiv:1301.6811 [hep-ph]]. [29] Stuart L. Shapiro, and Saul A. Teukolsky, Black Holes, White Dwarfs and Neutron Stars: The Physics of Compact Objects, Wiley, 2008.

Stable Bound States of Asymmetric Dark Matter Mark B. Wise and Yue Zhang

arXiv:1407.4121v1 [hep-ph] 15 Jul 2014

Walter Burke Institute for Theoretical Physics, California Institute of Technology, Pasadena, CA 91125 The simplest renormalizable effective field theories with asymmetric dark matter bound states contain two additional gauge singlet fields one being the dark matter and the other a mediator particle that the dark matter annihilates into. We examine the physics of one such model with a Dirac fermion as the dark matter. For a range of parameters the Yukawa coupling of the dark matter to the mediator gives rise to stable asymmetric dark matter bound states. We derive properties of the bound states including nuggets formed from N 1 dark matter particles. We also consider the formation of bound states in the early universe and direct detection of dark matter bound states. Many of our results also hold for symmetric dark matter.

I.

INTRODUCTION

An attractive idea for the origin of the cosmological dark matter (DM) density relies on a primordial DM asymmetry that prevents the near complete annihilation of the DM particles with their anti-particles resulting in the observed relic density. DM that arises from this mechanism is called asymmetric dark matter (ADM) [1]. If DM is asymmetric then the universe today is composed of DM particles with their antiparticles absent. An interesting possibility that can impact the properties of DM relevant for direct and indirect detections and influence its distribution in galaxies is the presence of stable bound states of DM particles. For ADM these bound states involve DM particles (but not DM anti-particles). The cosmology of ADM bound states have been explored previously in various contexts, such as the dark atom models with two species of ADM [2–5], and strongly interacting non-abelian hidden sector models [6–8]. In the Standard Model (SM) there is no particle that can be the DM. Minimality can be a useful guiding principle when considering speculative extensions of the SM. The purpose of this paper is to study one of the two minimal renormalizable extensions of the SM with stable ADM bound states. We will not discuss the mechanism that generates the primordial DM asymmetry. Rather we focus on the low energy effective theory well below the scale where the primordial asymmetry is generated. In the model we study, the DM is a Dirac fermion χ with no standard model quantum numbers. To facilitate annihilation of χ particles with their anti-particles χ ¯ in the early universe, we introduce a Yukawa-coupling of χ to a real scalar field φ lighter than the DM. Unlike U (1) gauge boson exchange1 , scalar exchange is always attractive among particles (or antiparticles), and so for a range of model parameters stable ADM bound states occur2 in the model we consider. The new scalar φ mixes with the Higgs boson, allowing it to decay to SM particles, and mediate interactions between ADM and SM particles that lead to a possible signal in the direct detection experiments. We discuss the spectrum of non-relativistic ADM bound states from two-body to multi-particle bound states, and the production of bound states in the early universe. We find regions of parameter space where most of the DM particles do not reside in bound states, and regions where most of the DM does. The border between these two regions occurs (roughly 1 2

DM interacting via dark U (1) gauge boson exchange was considered in for example Refs. [9, 10]. Having additional scalars aggravates the hierarchy problem of the standard model but in this case since the scalar is related to the DM density there may be environmental reasons for it being light compared with the Planck or GUT scales.

2 speaking) when the force mediator φ mass is comparable to the largest binding energy of the two body bound states.3 The low energy effective theory of ADM we consider must be (approximately) invariant under global U (1)χ transformations, χ → eiα χ. Like the ADM density the baryon density of the universe may arise from a primordial asymmetry, in that case in B − L. For the baryon asymmetry of the universe, the (approximate) conservation of B − L in the low energy effective theory, i.e., the SM, below the scale where the primordial asymmetry is generated is a consequence of gauge invariance and the particle content, since no renormalizable interactions between SM particles violate B − L.4 Naively this is not true for the ADM model we consider, global U (1)χ invariance is not an automatic consequence of the gauge symmetries and particle content. For example, while a Dirac mass term χχ ¯ preserves χ number a Majorana mass term χ ¯c χ does not. However, there could be an unbroken discrete gauge subgroup of U (1)χ that forbids the Majorana mass term resulting in a low energy effective theory, where the gauge symmetries are enough to ensure the renormalizable couplings in the low energy theory are invariant under U (1)χ . Even in models where the DM is not asymmetric, some of the results of the paper may be applicable. Scalar exchange could gives rise to stable bound states of DM particles and stable bound states of anti-DM particles. This will have an impact in cosmology and for direct detection. For a range of parameters, such bound states occur in for example [11–14]. II.

THE SIMPLEST FERMIONIC ADM EFT WITH BOUND STATES

The most minimal renormalizable effective theory for ADM contains complex scalar DM that annihilates via the Higgs portal in the early universe. However, it is not difficult to show that in this case there is not a robust region of parameter space where non-relativistic ADM bound states exist. Demanding the existence of stable ADM bound states in a renormalizable model requires one more degree of freedom than this minimal model. We focus on a renormalizable model with Dirac fermionic ADM χ and a real scalar mediator φ, 1 1 1 1 L = iχ ¯ 6 ∂χ − mχ χχ ¯ − χ(g ¯ χ + ig5 γ5 )χφ + (∂φ)2 − λ4φ φ4 − λ3φ φ3 − m2φ φ2 2 24 6 2 1 2 † 2 † 2 (1) − µφh φ(H H − v /2) − λφh φ (H H − v /2) − V (H) , 2 where H is the SM Higgs doublet, and V (H) is the usual Higgs √ potential. The vacuum expectation value for the neutral component of the Higgs doublet is v/ 2, and v = 246 GeV. We have shifted the scalar field φ so that it has no vacuum expectation value. After electroweak symmetry breaking, φ picks up a small mixing with the SM Higgs boson h, thus allowing it to decay into SM particles, and mediate the DM interaction to the SM. This simple model can already have a very complicated spectrum of bound states with important implications for cosmology and direct detection. With the above interactions, we calculate the cross section for χχ ¯ annihilation into the lighter mediator during thermal freeze out. In the region of parameter space with bound states mφ is much less than mχ . Neglecting mφ /mχ , µφh /mχ , λ3φ /mχ , and expanding to order v 2 the annihilation cross section is, 3π 2 2 2π π 2 2 hσvianni = α v + 2 αχ αI + α v , (2) 2m2 χ m 6m2 I 3

4

The other minimal renormalizable extension of the standard model with stable ADM bound states has scalar dark matter. Violation of this global symmetry occurs at dimension five through the operators that are responsible for neutrino masses.

3

m Χ = 1 TeV

m Χ = 100 GeV

1.000 0.500

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1.000 BBN&LUX excl. minimal model only

1.000 BBN&LUX excl. minimal model only

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10

10 keV

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3

FIG. 1: Ground state binding energy (in blue curves) of two-body DM bound state in the αχ − mφ space, for DM mass equal to 1 TeV (left), 100 GeV (middle) and 10 GeV (right), respectively. No two-body bound state exists in the dark blue region, and only one (1s) bound state exists in the lighter blue region. The yellow region is not consistent with BBN and LUX constraints in the minimal model described by Eq. (1). The black dashed curves are lower bound on αχ from having a large enough annihilation rate for the ADM, in the case αI = 0 (upper) and αI = αχ , respectively.

where v is the relative velocity between χ, χ, ¯ given by mχ v 2 ' 3T , and αχ ≡ gχ2 /(4π), αI ≡ g52 /(4π). We work to lowest order in perturbation theory neglecting the Sommerfeld enhancement factor which is not very important at freeze out [11, 15]. Asymmetric DM needs a larger annihilation rate than the WIMP, hσv(T ' mχ /26)ianni > 3 × 1026 cm3 /s [16]. With this condition, the symmetric component of DM can annihilate efficiently, and today’s relic density is dictated by the initial DM asymmetry. For the χχ ¯ annihilation, the contribution from the coupling gχ is velocity suppressed but not the contribution from interference between gχ and g5 . For non-relativistic interactions between DM particles the influence of g5 that is suppressed. In Fig. 1 the dashed lines are the limits on αχ assuming αI = 0 (upper dashed line) and αχ = αI (lower dashed lines) that arise from demanding that enough annihilations take place. In order to be consistent with Big-Bang Nucleosynthesis (BBN), φ has to decay before a second or so. In this model, φ decays to SM particles via mixing with the Higgs boson h, so its decays are similar to those of the Higgs boson [17]. For mφ below twice of the pion mass, its decay rate is !3/2 !3/2 G m2 m GF m2µ mφ 4m2µ 4m2e F e φ √ √ Γφ = 1− 2 Θ(mφ − 2me ) + 1− 2 Θ(mφ − 2mµ ) 4 2π mφ mφ 4 2π " #2 2 2 µ2 v 2 GF α2 m3φ 2 −1 φh √ + A1/2 + 3 A1/2 , (3) A1 + 2 × 3 3 3 3 m4h 128 2π where the second row is for φ → γγ decay and we have taken into account of the W -loop (A1 term) and the heavy quark loops (A1/2 term) from t, b, c, and A1 = −7, A1/2 = 4/3 [18]. The φ → µ+ µ− decay dominates above the two-muon threshold. For heavier φ, hadronic decay channels open, and the dimuon branching ratio reduces to around 10% [19], for φ mass up to a few GeV. The φ − h mixing can also mediate the interaction between DM and SM particles, and lead to

4

2 ´ 10-5

@ΜΦh vmh 2 DMax

1 ´ 10-5 5 ´ 10-6 2 ´ 10-6 1 ´ 10-6 5 ´ 10-7 2 ´ 10-7

10

20

50

100 200 m Χ HGeVL

500 1000 2000

FIG. 2: Upper bound on the φ − h mixing parameter µφh v/m2h from dark matter direct detection (LUX), assuming all DM are free particles today. The solid and dashed curves corresponds to αχ = 0.1 and 0.01, respectively. We fix mφ = 500 MeV.

DM direct detection signals. The direct detection cross section for χ scattering on a nucleon is [20] σSI =

4αχ f 2 m4N µ2φh m4h m4φ

,

(4)

assuming m2φ |q 2 | where q is the momentum transfer, and f ' 0.35 [21]. Fig. 2 shows the constraints on the φ − h mixing from the DM direct detection by the LUX collaboration [22] for given φ mass and αχ . We find that combining the LUX and BBN constraints generically requires the φ → µ+ µ− decay channel to be open, i.e., mφ & 210 MeV. The regions of parameter space that are excluded by direct detection and consistent with BBN are shaded yellow in Fig. 1. This eliminates a large part of the available parameter space. More complicated models with an enlarged dark sector may evade this constraint. See the appendix for an example. A.

Two-body Bound States

The exchange of a scalar field φ gives an attractive force among the ADM particles. For sufficiently light φ, most of the ADM in the universe could reside in bound states. Because the φ force is always attractive, multi-particle ADM bound states (N ≥ 2) are expected. In this section, we discuss two-body bound states. The states containing more than two DM will be discussed in the following subsection. For two-DM-particle system, the non-relativistic Hamiltonian describing two DM interaction is H = −

∇2cm ∇2 αχ −mφ r − − e + Hint , 4mχ mχ r

(5)

where xcm is the center of mass coordinate, and r is the relative position of two χ’s inside the bound state. Hint is the interaction for on-shell φ creation/annihilation, which controls transition rates, that involve φ particle emission and absorption. The other terms in Eq. (5) control the spectra and wave functions of the two-body bound states in the non-relativistic limit. We restrict our attention

5 to the region of parameter space where αχ . 0.1 which insures that the two-particle bound states are non-relativistic. The non-relativistic two-body bound state problem with a Yukawa potential has been solved numerically in [23] (see also [24]). In order for a bound state to exist, the screening length (∼ 1/mφ ) must be large enough compared to the size of the corresponding wave function (∼ the Bohr radius). The condition for having at least one bound state (the 1s state) is αχ mχ > 0.8399 . 2mφ

(6)

In Fig. 1, we plotted contours of fixed 1s state binding energy as thin blue curves in the αχ versus mφ plane, for fixed DM mass in each panel. The dark blue region does not satisfy the bound state condition Eq. (6) because φ is too heavy. In the light blue region, 1s state is the only existing bound state. Outside these regions, for mφ αχ mχ , the φ force is very close to Coulombic, and the solution approaches to the hydrogen-like state, with a Bohr radius a0 =

2 . αχ mχ

(7)

In this case, the ground state wave function and binding energy are, approximately, 1 −3/2 ψ0 (r) = √ a0 e−r/a0 , π

BE 0 =

αχ2 mχ . 4

(8)

A two-particle bound state Bi can decay to one with greater binding energy Bf by real or virtual φ emission. Real φ emission is kinematically allowed when the the difference in binding energies ∆BE between the final and initial states is greater than the mass of the φ. Also bound states can be formed by scattering two χ’s and emitting either a real or virtual φ and dissociated by scattering the bound state with a φ. The couplings of the φ to the SM particles are restricted to be very small by current direct detection bounds so we concentrate on real φ emission and absorption in these processes. Also for simplicity we only consider the s-wave bound states. They have the spins of the χ’s combined into a total spin zero state. The interaction Hamiltonian that is needed for these calculations is h r r i Hint = gr φ xcm + + φ xcm − . (9) 2 2 Our calculations are valid as long as the χ’s are non-relativistic but the φ can be relativistic. Excited bound states can decay to lower bound states by φ emission, Γ(Bi → Bf + φ) = 8αχ k|Gif (k)|2 ,

(10) q where k = ∆BE 2 − m2φ is the magnitude of the φ momentum and the transition form factor is Z Gif (k) =

d3 re−k·r/2 ψf∗ (r)ψi (r) .

(11)

We will need, for the cosmology discussion in the forthcoming section, the cross sections for the formation and dissociation of the two-body states via real φ emission and absorption (see Fig. 3). We first consider the formation process in the center of mass frame, χ(p) + χ(−p) → Bi + φ. The cross section is expressed in terms of the form factor. Z Fi (p, k) = d3 re−ip·r ψi (r)(eik·r/2 + e−ik·r/2 ) , (12)

6

B

B

FIG. 3: Feynman diagrams for two-body ADM bound state formation (dissociation) with an on-shell φ emission (absorption).

where ψi is the spatial wave function for the i-th s-wave bound state and k is the three momentum of the outgoing φ. In the cases considered here, we can use the dipole approximation, neglecting the φ three momentum k. In the case of the ground state i = 0 approximating the wave function by the Coulombic one in Eq. (8), F0 (p, 0) =

√ 3 16 πa02 1 + a20 |p|2

2 .

(13)

Using the dipole approximation, in the center of mass frame, σ(χ(p) + χ(−p) → Bi + φ)v =

αχ k|Fi (p, 0)|2 , 2

where energy conservation fixes the magnitude of the final state φ momentum to be, q k = (BEi + p2 /mχ )2 − m2φ .

(14)

(15)

It is evident from the above formula that as long as the χ’s are non-relativistic, the dipole approximation is reasonable. Similarly the cross section for the dissociation process in the lab frame, Bi (0) + φ(k) → χ + χ is, σ(Bi (0) + φ(k) → χ + χ)v =

αχ mχ p|Fi (p, 0)|2 , Ek

(16)

where p is the magnitude of the relative momentum of the final stateqχ’s, i.e. p = (p1 − p2 )/2 with

p1 and p2 are the three momenta of the two final state χ’s. Ek = k 2 + m2φ is the energy of the incoming φ. Energy conservation determines the magnitude of the relative χ momentum to be, q p = mχ (Ek − BEi ) , (17) where BEi is the binding energy of the two body bound state Bi . If some of the DM is in the two-body bound ground state today the direct detection predictions may be modified. The momentum transfer associated with direct detection is less than 1/a0 so under most circumstances the scattering is coherent, the form factor suppression negligible and the cross section is just 4 times the single χ cross section in Eq. (4). An exception occurs when the two body scattering length is very large. At zero momentum the (spin zero) χ − χ scattering cross section σ(χχ → χχ) = 4πa2 , where a is the spin zero scattering length. If there is a two body bound state (or resonance) very near threashold (i.e. zero binding energy) then the scattering

7 length a is very large5 . In this case there are universal properties at low momentum for few body χ systems. This is familiar from the effective range expansion for two body nuclear physics processes (e.g. n + p → d + γ) and was applied to DM direct detection in [25]. B.

Dark Matter Nuggets

As mentioned above, the φ force among DM particles is always attractive. Therefore, for small enough φ mass we expect DM to have N > 2 particle bound states (nuggets). For sufficiently large N , we assume the DM nugget can be described as degenerate Fermi gas. Undoubtably this is a gross simplification of the dynamics. One expects more complicated phenomena like pair formation [26] to occur and impact the equation of state for the DM. However, our main purpose here is not a quantitative analysis of the spectrum and properties of multi-particle DM bound states but rather to argue that such states exist and that they are probably small enough that in direct detection experiments the scattering is coherent and gives rise to a cross section that grows as N 2 . This will also affect the capture rate for DM by neutron stars [27, 28]. We first discuss this multi-particle bound state problem using a heuristic approach where the DM density is constant and then a more quantitative approach that relies on hydrostatic equilibrium. In expressions for physical quantities the scaling with N , mχ and αχ is the same in the heuristic and hydrostatic equilibrium approaches. We restrict our attention to the non-relativistic regime so we can neglect relativistic corrections to the potential energy, for example from the Darwin term. Also we will find that the size of the state decreases with N so that we can neglect mφ replacing the Yukawa potential by a Coulomb potential. 1.

Heuristic Approach

Assuming a constant density of χ particles filling the momentum levels up to the Fermi momentum pF the number density of χ particles is n=

p3F . 3π 2

(18)

For a spherical volume of radius R, the total number of particles is N = 4p3F R3 /(9π). The kinetic energy KE, and potential energy P E, of the χ particles expressed as a function or the total number of particles N and the radius R are 5/3 3N 2 αχ 2π 2/3 5/3 9 KE = N , P E = − . (19) 15R2 mχ 4 5R Minimizing the total energy E = KE + P E with respect to R at fixed N determines the radius to be, 2/3 1 9π R= . (20) 1/3 4 N αχ mχ Notice that the volume of the nugget decreases as 1/N and if mφ a0 1, then we also have for large N that mφ R 1. Hence as long as two body bound states exist the Yukawa potential can be treated as Coulombic for nuggets. 5

This is the origin of the enhancements in [11], [13] and [14].

8 At the value of R in Eq. (20) the kinetic and potential energies become, KE ' 0.08 αχ2 mχ N 7/3 , P E ' −0.16 αχ2 mχ N 7/3 .

(21)

For large N the nuggets have a binding energy that is of order N 4/3 times the binding energy of N/2 two body bound states BE0 . There are a number of conditions that must be satisfied to apply even the crude approximations we have made. Firstly as N increases the Fermi momentum increases and the system eventually becomes relativistic. Demanding that pF /mχ 1 implies that N

9π 4

1/2

αχ−3/2

' 2.7

1 αχ

3/2 .

(22)

Determining the properties of the nugget using partly classical methods will be valid if pF R 1. This implies that −1/3 9π ' 0.52 . (23) N 4 In the presence of a background number density for χ’s, the Yukawa coupling of the scalar induces a density dependent tadpole and for large enough R a scalar expectation value. However, neglecting mφ and λ3φ and treating λ4φ as order unity, p we find that these effects are subdominant compared to those we have included, provided N 1/αχ . 2.

Hydrostatic Equilibrium

For a degenerate Fermi gas the equation of state relating the pressure density p to the number density n is p = Kn5/3 ,

(24)

where K = 5−1 32/3 π 4/3 m−1 χ , and n is the number density of χ particles. For a stable solution, the Fermi pressure is balanced by the attractive φ force among the particles. When 1/mφ is larger than the size of the nugget, the attractive force is Coulomb-like. In this case, the hydrostatic equilibrium equation is 1 d r2 dp = −4παχ n . (25) r2 dr n dr Together with the above equation of state, Eq. (24), this equation can be solved for N DM particles and it has a finite-size solution, with R=

(3π)2/3 2−7/3 2 0 4.5 (ξ1 |θ (ξ1 )|)1/3 · ξ1 ' 1/3 , 1/3 N αχ mχ N αχ mχ

(26)

where θ(ξ) is the solution to the Lane-Emden equation with index n = 3/2, and ξ1 = 3.65, ξ12 |θ0 (ξ1 )| = 2.71 [29] . The Fermi momentum near the center of the nugget is (pF )c =

22/3 N 2/3 αχ mχ ' 0.1N 2/3 αχ mχ . 31/3 π 4/3 (ξ12 |θ0 (ξ1 )|)2/3

For this description to apply, there are consistency conditions.

(27)

9 • Non-relativistic condition: (pF )c mχ requires N (0.1αχ )−3/2 . For αχ ≤ 0.1, the right hand side is of order 103 or larger. • Classical description: (pF )c R 1 requires N 6.4 . • Long range force condition: mφ R 1 requires mφ N 1/3 αχ mχ /4.5 . For large N nuggets are smaller than the the two-body bound ground state, while the momentum of DM inside the nugget is much larger than the two-body case. Therefore, as long as the two-body bound state exists, the screening effect due to mφ can be neglected. Within the degenerate Fermi gas picture, for very large N , a non-relativistic description is no longer valid. Other interactions we have not included become important in the analysis of such systems. However, it seems plausible that relativistic bound states exist, and even black holes. If most of the DM ends up in black holes, then they must be cosmologically stable. As we noted before, since the size of DM nuggets shrinks with N we expect their direct detection scattering cross section to be coherent and be proportional to N 2 . III.

COSMOLOGY

In this section, we study the formation of bound states in the early universe. In general, there are two stages when bound states may form most efficiently: 1) shortly after the DM freeze out when the ADM number density is still high, 2) at a later stage where structure growth has gone non-linear and the DM density can be locally large. We first perform a calculation of two-body bound state formation in stage 1), taking into account two competing processes, formation χχ → Bφ, and dissociation Bφ → χχ. The fraction of DM in bound states depends on the interplay between these two rates and the Hubble parameter. Our goal here is to understand two-body bound state formation. To calculate the formation of bound states with more than two particles, we need to know the binding energies and wave functions of those states. For convenience, in this section we fix µφh v/m2h = 10−6 and take mφ > 2mµ and less than a few GeV. These values are consistent with constraints from DM direct detection. In this region of parameter space, φ decay is dominated by hadronic and the two muon-final state. Neglecting threshold effects, the two-muon contribution gives the bound, 1 GeV −4 τφ . 10 sec . (28) mφ Recall that then the universe is 10−4 second old, its temperature is ∼ 100 MeV. When t < 1/τφ in the early universe there is a plasma of φ particles which couples to the DM. In order to calculate the averaged bound state formation and dissociation rates in this plasma, we need to know the energy/momentum distributions of χ and φ. At very high temperature, the mediator φ was in thermal equilibrium with SM fermions via the Higgs boson exchange, φφ ↔ f f¯. For λφh ∼ O(1) and mφ . 1 GeV, such interactions freeze out at temperature equal to 1 GeV or so, slightly below the charm quark threshold. Afterwards, φ can only remain in chemical equilibrium with itself through the 2 ↔ 3 scattering φφ ↔ φφφ, with the λ3φ , λ4φ couplings. This allows it to have its own temperature Tφ , which satisfies Tφ ' Tγ

1, [g∗ (Tγ )/g∗ (1 GeV)]1/3 ,

Tγ > 1 GeV Tγ < 1 GeV

(29)

10 where Tγ is the photon temperature and g∗ (Tγ ) is the number of relativistic degrees of freedom in the SM at Tγ . When the temperature falls below mφ , the φ number density becomes Boltzmann suppressed and the 2 ↔ 3 scattering process freezes out, at temperature around an order of magnitude below mφ . Another important way to deplete the φ’s is decay. For simplicity, we restrict ourselves to the region of parameter space where φ decay happens before or around the freeze out, and in this case the phase-space distribution of φ is fφ (E) ' e−E/Tφ e−1/(2Hτφ ) .

(30)

Second, after the anti-DM χ ¯ are efficiently depleted (T . mχ /30), the remaining ADM component χ can stay in kinetic equilibrium with φ, via the elastic scattering χφ → χφ. For the range of parameters we study, this rate is always larger than the Hubble rate, until φ’s decay away. In this case, the phase space distribution of DM χ is Tγ3 ΩDM mp −p2 /(2mχ Tφ ) 5/2 −1/2 η fχ (p) = Ce , C=2 π ζ(3) , (31) Ωb mχ (mχ Tφ )3/2 where η ' 6 × 10−10 is the ratio of baryon to photon number in the universe, and mp is the mass of the proton. We will use the φ and χ distributions described above to calculate the thermal averaged rates. For the formation rate Γf orm per particle, we do a thermal integral of the cross section Eq. (14) over the incoming χ momentum. The thermally averaged dissociation rate Γdiss per particle is obtained by integrating Eq. (16) over the incoming φ energy. When both Γf orm and Γdiss are larger than the P Hubble rate and only two-body bound states exist, the dark ionization fraction, Xd ≡ nχ /(2 i nBi + nχ ), (the index i goes over all possible two-body bound states) satisfies the dark sector counterpart of the Saha equation6 1 − Xd 8ζ(3) = √ η 2 π Xd

5.4 GeV mχ

Tφ mχ

3/2 X

eBEi /T .

(32)

i

The large number of φ’s in the plasma implies that Γdiss Γf orm until the time of φ decay. In the end, how much two-body bound states are formed is determined by the comparison of formation rate and Hubble rate at that time. For simplicity below, we only discuss the formation and dissociation of the ground state. However, kinematically, excited states are harder to form and easier to destroy. Therefore, we expect most of the two-body bound state formation to occur in the ground state. The parameter space can be divided into two regimes. • Case A: mφ BE0 . In this case, the mediator mass is much larger than the binding energy, and the formation process χχ → Bφ cannot happen unless the two DM are energetic enough. In other words, for DM in kinetic equilibrium, the temperature of the universe must be large enough. When the temperature falls below a threshold Tth = (mφ − BE0 )/3 ' mφ /3, the formation rate becomes exponentially suppressed. In contrast, with a plasma of φ the dissociation rate does not shut off until φ eventually decays, which happens at a temperature much lower than Tth . This feature is shown as the left panel of Fig. 4, where when Γf orm falls below H, Γdiss is still much larger than the Hubble rate H. In this case, any bound states that were formed will 6

A similar equation could be derived in the nugget case.

11 Case B: m Χ = 1 TeV, Α Χ =0.1, mΦ =300 MeV 1012

104

104

10-4

10-4

-12

10

GΦB® ΧΧ

Rates HGeVL

Rates HGeVL

Case A: m Χ = 100 GeV, Α Χ =0.05, mΦ =300 MeV 1012

G ΧΧ®BΦ

10-20

H

-28

10

10-36 m Χ 26

10-12

G ΧΧ®BΦ

10-20

H

10 mΦ

10 MeV

1 MeV

GΦB® ΧΧ

-28

10-36 m Χ 26

mΦ

T

10 MeV

1 MeV

T

FIG. 4: Temperature dependence of the two-body ADM bound state formation (red), dissociation (blue) rates in together with the Hubble expansion rate (black). In both cases, we set µφh v/m2h = 10−6 such that both DM direct detection and BBN bounds can be satisfied. m Χ = 200 GeV, Α Χ =0.08, mΦ =500 MeV 10

12

Rates HGeVL

104

GΦB® ΧΧ

10-4 10-12

G ΧΧ®BΦ

10-20 10

H

-28

10-36 m Χ 26

mΦ

10 MeV

1 MeV

T

FIG. 5: Same as Fig. 4, but the parameters are chosen such that the critical condition discussed above Eq. (35) is satisfied.

eventually be ionized back to unbounded DM particles. For the same reason, bound state formation when the structure growth becomes non-linear is also suppressed because the DM is more non-relativistic. • Case B: mφ BE0 . In this case, the binding energy release itself is sufficient to produce an on-shell φ. There is no temperature threshold for bound state formation. The thermally averaged formation cross section Eq. (14) can be simplified in the following two regimes 4παχ6 mχ /(27Tφ3 ), Tφ BE0 hσ(χ + χ → B0 + φ)vi ' (33) 2 256π/mχ , Tφ BE0 Because nχ ∼ Tφ3 , the formation rate remains constant when Tφ BE0 , and scale as ∼ Tφ3 (decreases faster than Hubble rate) when Tφ BE0 . Hence there is a lower bound on the

12

m Χ = 1 TeV

m Χ = 100 GeV

1.000

m Χ = 10 GeV

1.000

0.500

BBN&LUX

m

E0 =B Φ

1.000

0.500

BBN&LUX

m

E0 =B Φ

0

BBN&LUX 0.100

0.050

0.050 ΑΧ

0.100

0.050 ΑΧ

0.100 ΑΧ

E =B mΦ

0.500

0.010

0.010

0.005

0.005

0.010

No bound state

0.005

No bound state No bound state 0.001 0.001

0.01

0.1 1 mΦ HGeVL

10

100

0.001 0.001

0.01

0.1 mΦ HGeVL

1

10

0.001 0.001

0.01

0.1 mΦ HGeVL

1

3

FIG. 6: Same parameter space as Fig. 1. The thick red curves represent where the critical condition like Fig. 5 is reached. The dot-dashed lines are where mφ = BE0 is satisfied.

coupling constant αχ for the formation rate to ever be larger than the Hubble rate mχ mχ ' 5 , αχ & p 10 GeV ηMpl mp

(34)

where Mpl = 1.2 × 1019 GeV is the Planck scale. If this condition is satisfied, the formation rate can remain greater than the Hubble rate for a long time. In contrast, there will be a threshold for dissociation. At mφ < T < EB , there is a suppression in the number of φ in the plasma that are energetic enough to ionize the bound state. Moreover, there is a sharper suppression when φ begins to decay. This allows us to have a picture where the ground state formation process is still active (Γf orm > H) when dissociation is suppressed (Γdiss < H), as shown in the right panel of Fig. 4. In this regime, the two-body ground state can efficiently form. In this regime, the formation of more than two-body bound states is also expected to be efficient. • The critical case. In the region that interpolates between the two above limiting cases, for given mφ , we find it is always possible to arrange the parameters such that both formation and dissociation freeze out at similar temperature Tc , where Γdiss (Tc ) = H(Tc ), but Γf orm (Tc ) is somewhat below H(Tc ), as shown in Fig. 5. Below Tc , the dissociation rate is more suppressed because of φ decay. In this case, the fraction of DM particles that finally end up in the two-body ground state is roughly, P '

Γf orm (Tc ) . H

(35)

Since P is smaller than one, we expect the formation of more than two-body bound states to be further suppressed. We summarize the results in the αχ versus mφ parameter space in Fig. 6. The thick red curves are where the above critical condition is satisfied. Below the thick red curves, almost all the DM ends up as unbound χ particles. Above thick red curves, most of the DM resides in bound states. These curves lie at mφ somewhat above the binding energy BE0 because of the kinetic energy of the χ particles (see Eq. (15)). Note the thick red curves bend up towards the mφ = BE0 line near

13 the two muon threshold for φ decay. This occurs because the φ decay rate is suppressed which postpones the suppression of dissociation rate, and a larger value of αχ is needed to reach the critical point. The force between any pair of DM particles is attractive and so for mφ a0 1, the binding energy for a bound state with N particles grows faster than linearly with N . That was what we found in the degenerate Fermi gas model where the binding energy grew as N 7/3 . Thus it is likely that for mφ < BE0 there are no thresholds that suppress the formation of non-relativistic bound states with more than two particles. IV.

CONCLUDING REMARKS

We showed that, for a range of parameters, one of the simplest effective theories of asymmetric dark matter has a rich spectrum of bound states. In this paper, we explored some of the features of the spectrum, and the implications for cosmology and dark matter direct detection. We find a region of parameter space where the dark matter in the universe is primarily in bound states. Roughly speaking, this occurs when the binding energy of the two-particle ground state is greater than the mediator mass. However, there are a number of issues that require further examination. For example, the details of the spectrum of the bound states with more than two dark matter particles, i.e., nuggets, and the formation of these multi-particle bound states in the early universe. Without further investigation of these issues, it is even conceivable that for a range of parameters most of the asymmetric dark matter ends up as black holes. In that case, the black holes must have a lifetime longer than the age of the universe. It seems worthwhile to elucidate further the bound state properties and cosmology in this simple model for asymmetric dark matter. Some of the work in this paper is also applicable to dark matter that is not asymmetric. For a range of parameters, scalar exchange could gives rise to stable bound states of dark matter particles and stable bound states of anti dark matter particles. One difference from the asymmetric case is that the values of αχ and αI are constrained to give the correct dark matter relic density. Acknowledgement

We would like to thank Sean Carroll, Clifford Cheung, Francis-Yan Cyr-Racine, Michael RamseyMusolf, David Sanford, Philip Schuster and Natalia Toro for useful discussions and comments on a draft. This work is supported by the Gordon and Betty Moore Foundation through Grant No. 776 to the Caltech Moore Center for Theoretical Cosmology and Physics, and by the DOE Grant DE-SC0011632, and also by a DOE Early Career Award under Grant No. DE-SC0010255. APPENDIX: A MORE NATURAL MODEL

The smallness of µφh in the minimal we consider requires an awkward fine-tuning that is unlikely to have an environment origin. More complicated models can avoid this feature. For example, suppose the dark sector possesses a global dark isospin symmetry SU (2), under which the DM χ is a doublet and the mediator φ is a triplet. The Lagrangian is 1 1 1 1 L = iχ ¯ 6 ∂χ − mχ χχ ¯ − gχ χΦχ ¯ + Tr(∂Φ)2 − λ4φ TrΦ4 − λ3φ TrΦ3 − m2φ TrΦ2 4 24 6 2 1 − λφh TrΦ2 (H † H − v 2 /2) − V (H) , 2

(36)

14 where Φ = σ a φa . In order for the φ to decay, a dark doublet of left-handed fermions ψ is introduced that couples to Φ via the interaction ψ¯c Φψ + h.c.. Note the dark isospin forbids a mass term for ψ. The coupling between Φ and ψ should be large enough for Φ to decay before BBN. In this model, the DM direct detection occurs at the one loop level (see Fig. 7). The cross section is given by Eq. (4), with µφh /m2φ replaced by ∼ gχ λφh /(16π 2 mχ ), which is adequately small even for λφh of order unity.

v

p

h p

FIG. 7: Feynman diagram for direct detection in the SU (2) model.

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